The first part of my talk will discuss ways of establishing cubic lifespan bounds for quasi-linear dispersive equations via the \emph{modified energy method}. This robust method is an upgrade of the normal form energy method introduced by Shatah in 1983 for the Klein-Gordon equation. For simplicity we will first implement it for the Burger's-Hilbert equation (which is not dispersive), but where it nevertheless works. The second part of my talk will discuss global existence of solutions provided that the initial data is small and spatially localized via the \emph{testing with wave packets method}. We will show how it works for one simple example: for the one dimensional cubic NLS.